Help me understand Riemannian Manifolds I understand differentiable manifolds and the need of them. Thanks to the atlas structure, we can develop a differential calculus on spaces that look locally like $\mathbb R^n$.
Now, what is the need of introducing a metric on the tangent spaces of a smooth manifold (that is what is called a Riemannian metric, and a smooth manifold equipped with a Riemannian metric is called a Riemannian manifold) ? What does it allow us to do and why would we want to do that ?
I feel everything that can be done in $\mathbb R^n$ can already be done in a smooth manifold by using coordinates charts. So what is the need of this extra Riemannian structure ? What is the difference between a (smooth) metric space and a Riemannian manifold ? Why do we equip a metric on the tangent spaces and not on the space itself ? What is the intuition behind this structure ?
Maybe those are very broad questions, if needed, you can restrict to what really interest me: geodesics, ie the shortest continuous way of going from a point A to a point B. When I read about geodesics, it's all about Riemannian manifolds. But I do not understand why ? Geodesics need only a notion of distance: why not work in metric spaces then ? Are Riemannian manifolds a subset of metric spaces ?
 A: If we want to talk about geometry on a smooth manifold $M$, we need an additional structure allowing to determine

*

*the length of a curve


*the angle between two curves at a point of intersection.
Although both concepts are well-defined for curves in $\mathbb R^n$ (or more generally for curves in an open $V \subset \mathbb R^n$), it is impossible to transfer them via charts to smooth manifolds. This comes from the fact that the transition functions between charts in general do not preserve length and angles.
You are right that the length of a curve can be defined based on a metric on $M$ (see for example here), but this is impossible for angles between two curves.
Given two curves $\gamma_i : (-a_i,a_i) \to \mathbb R^n$ intersecting at $t=0$, the angle between $\gamma_1, \gamma_2$ is usually defined as the angle $\alpha$ between the tangent vectors $v_1 = \gamma'_1(0), v_2 = \gamma'_2(0)$, and this is determined via the standard inner product on $\mathbb R^n$:
$$ \cos \alpha = \dfrac{\langle v_1,v_2 \rangle}{\lVert v_1 \rVert \lVert v_2 \rVert} \tag {1}$$
The length of a curve $\gamma : [a,b] \to \mathbb R^n$ is given by
$$\int_a^b \lVert \gamma'(t) \rVert dt = \int_a^b \lVert v(t) \rVert dt \tag{2}$$
where $v(t) = \gamma'(t)$ is the tangent vector ("speed vector") at $\gamma$ at time $t$.
This can be taken as the definition of the length for smooth curves or as theorem if one works with the more general concept of rectifiable curves.
In a smooth manifold tangent vectors at $p  \in M$ can be introduced as equivalence classes of curves through $p$.
Thus, given curves $\gamma_i : (-a_i,a_i) \to M$ intersecting at $t=0$ in $p$, the angle  $\alpha$ between $\gamma_1, \gamma_2$ should be defined as above based on an inner product on $T_pM$: In fact, the $\gamma_i$ represent tangent vectors $v_i = [\gamma_i] \in T_p M$ which allows use formula $(1)$ also for a general $M$.
The length of a curve $\gamma : [a,b] \to M$ is given by $(2)$ in the form
$$\int_a^b \lVert v(t) \rVert_{\gamma(t)} dt$$
where $\lVert - \rVert_p$ is the norm induced by the inner product on $T_pM$ and $v(t)$ is the tangent vector at $\gamma$ at time $t$ which is represented by the curce $\gamma_t : [a - t, b- t] \to M, \gamma_t(s) = \gamma(s+t)$. Note that if $t \in [a,b]$, then $0 \in [a -t, b-t]$ and $\gamma_t(0) = \gamma(t)$
Now you should see why the additional structure of inner products on all tangent spaces $T_pM$ (i.e. a Riemannian metric) is needed for geometry on $M$.
